View source for Primary ideal

''of a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744601.png" />''

An [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744602.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744604.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744605.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744606.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744607.png" />. In the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744608.png" /> of integers a primary ideal is an ideal of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p0744609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446010.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446011.png" /> is a natural number. In commutative algebra an important role is played by the representation of an arbitrary ideal of a commutative [[Noetherian ring|Noetherian ring]] as an intersection of a finite number of primary ideals — a [[Primary decomposition|primary decomposition]]. More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446012.png" /> denote the set of prime ideals of a [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446013.png" /> that are annihilators of non-zero submodules of a [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446014.png" />. A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446015.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446016.png" /> over a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446017.png" /> is called primary if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446018.png" /> is a one-element set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446019.png" /> is commutative, then every proper submodule of a Noetherian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446020.png" />-module that cannot be represented as an intersection of submodules strictly containing it is primary. In the non-commutative case this is not true and therefore attempts have been undertaken to construct various non-commutative generalizations of the notion of primarity. E.g., a proper submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446021.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446022.png" /> is called primary if for every non-zero injective submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446023.png" /> of the [[injective hull]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446024.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446025.png" /> (cf. [[Injective module|Injective module]]) the intersection of the kernels of the homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446027.png" /> is trivial. Another successful generalization is the notion of a tertiary ideal [[#References|[4]]]: A left ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446028.png" /> of a left Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446029.png" /> is called tertiary if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446031.png" />, it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446032.png" /> that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446033.png" />, there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446035.png" />. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446036.png" /> is primary if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446037.png" /> satisfies the Artin–Rees condition: For arbitrary left ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446039.png" /> there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074460/p07446041.png" /> (cf. [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Goldman,   "Rings and modules of quotients"  ''J. of Algebra'' , '''13''' :  1  (1969)  pp. 10–47</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Lesieur,   R. Croisot,   "Algèbre noethérienne noncommutative" , Gauthier-Villars  (1963)</TD></TR></table>